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There are much-discussed theorems like Fermat's theorem on sums of two squares which make statements about odd primes only. This makes $2$ seem to be a "special" prime. In their book The book of numbers, Conway and Guy accordingly state that "Two is celebrated as the only even prime, which in some sense makes it the oddest prime of all."

On the other hand, the fact that $2$ is the only even prime is completely trivial, because the term "even" means the same thing as "divisible by $2$" and every prime number has the property that it is the only prime which is divisiable by itself.

So my question is: Is there really something special about even primes and if yes, what is it? Does aesthetics with regard to the theorems we are looking for play a role or is there a mathematical reason? Do we have theorems about primes which aren't divisible by $3, 5, ... $ or are there only results which don't apply to even primes?

Edit: As the user A.G has mentioned in a comment below, in many cases where we have a regular pattern, the fact that $2$ is too small for the pattern to kick in yet seems to be the decisive thing. So in these cases, the notable thing is not that $2$ is the only even prime but that it is the smallest prime.

Marc
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  • If $n$ is even, then $n=2k$ for some $k\in\Bbb Z$. But now, by definition, $2\mid n$, so either $n=2$ or $n$ is not prime (but not both). – Shaun Feb 15 '20 at 15:57
  • I don't see how this makes $2$ notable. If you replace $2$ by an arbitrary prime $p$ and "even" by "divisible by $p$" your statement is still true. – Marc Feb 15 '20 at 16:00
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    What makes $2$ seem special is that English has a well known word for divisible by $2$ but not $3, 5, ...$. – badjohn Feb 15 '20 at 16:06
  • Here is a nice fact about even numbers: Def: A binary block code with $m$ codewords, each of size $n$ bits, is a list of $m$ distinct $n$-bit binary strings. Consider such codes with $m\geq 3$. It is possible to design such a code that can detect all odd-number of errors (just use a single parity check bit), but impossible to design one that can detect all even-number of errors. (An "error" is a bit flip.) An example parity check code with $n=3$ and $m=4$ (that can detect all odd-number of errors) is $$ {(0,0,0), (0,1,1), (1,0,1), (1,1,0)}$$ – Michael Feb 15 '20 at 16:57
  • What I meant is that you only use 0 and 1 (two elements). If you use three elements in the definition the answer would be different. – A.G Feb 15 '20 at 17:02
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    There is only a pair of consecutive prime numbers, the first one of them being 2. It's not too much, but for the moment I am not aware of any other thing. Nice question. – A.G Feb 15 '20 at 17:20
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    Distinguishing even from odd numbers is important in many aspects often with nothing to do with divisibility. It's basically a toggle and, for example, boolean algebra is very significant and important whereas an other divisibility...meh, it's covered in modular arithmetic as a whole. That with only one exception all primes have the same parity is significant. – fleablood Feb 15 '20 at 18:25
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    This discussion in MathOverflow seems related. – pregunton Feb 15 '20 at 18:33
  • @A.G "There is only a pair of consecutive prime numbers, the first one of them being 2". The number 3 has a similar property: (3,5,7) is the unique (etc.) of three "consecutive" prime numbers (where now "consecutive" has the obvious meaning). So my example above is not valid. – A.G Feb 15 '20 at 18:40
  • All $51$ perfect numbers found so far are divisible by $2$; all of these only have two distinct prime factors, and the other one only divides one of these perfect numbers. Possibly this is true for all perfect numbers, but if not then any perfect number not divisible by $2$ would have at least ten distinct prime factors. – Henry Feb 16 '20 at 02:27
  • Framing challenge: "even prime s"?!? Ignoring associates (as one does...), what's another one? – Eric Towers Feb 16 '20 at 04:23
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    See here for a lot of related discussion. – Jyrki Lahtonen Feb 16 '20 at 08:10
  • Speaking of perfect numbers, the Euclid-Euler theorem states that all even perfect numbers have the form $2^{p-1}(2^p-1)$ where $2^p-1$ is a (Mersenne) prime. Obviously, there are no Mersenne-like primes of the form $q^n-1$ for $q>2$ because $q-1|q^n-1$. I suppose there are other situations where 2 is weird because $\gcd(p-1, m)=1$ for all $m\ge1$ only when $p=2$... – PM 2Ring Feb 16 '20 at 09:01
  • MathOverflow: What is special about the prime numbers $2$ and $3$? – Torsten Schoeneberg Feb 18 '20 at 17:42

5 Answers5

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The quip about "2 being the only even prime" is a bit silly, as you say, since 3 is the only divisible-by-3 prime, etc. For that quip, it's just that parity (odd-or-even) exists in the ambient language.

For $p$ prime, the $p$th roots of unity are in $\mathbb Q$ for $p=1$. Similarly, the $p$th roots of unity lie in all finite fields (of characteristic not $p$...) only for $p=2$.

Quadratic forms and bilinear forms behave differently in characteristic two.

Groups $SL(n,\mathbb F_q)$ do not assume their general pattern yet for small $n$ and $q=2$.

The index of alternating groups in symmetric groups is $2$.

Subgroups of index $2$ are normal.

The canonical anti-involution on a non-commutative ring, that reverse the order of multiplication, is of order $2$.

paul garrett
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    Some of these examples use also the cloice of 2 (quadratic forms, alternating, the (binary) multiplication on a ring). Some others (roots of unity, normal subgroup) are due to the fact that 2 is a small number (the fact of being prime plays no role). – A.G Feb 15 '20 at 17:14
  • Thanks for pointing towards characteristic $2$. Although my abstract algebra is a bit rusty, I remember wondering why characteristic $2$ was mentioned as an exception there so often. – Marc Feb 15 '20 at 17:18
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    But for instance in elliptic curves (degree 3 ... ) the exceptions are in characteristic 3 ( and 2). So I think it is nothing special of the prime 2. – A.G Feb 15 '20 at 17:22
  • @A.G, so how can we explain that, by coincidence, 2 is the smallest prime? :) That is, seriously, it has both size and divisibility anomalies. – paul garrett Feb 15 '20 at 17:43
  • @A.G: What about curves of degree 5, in which characteristics are the exceptions there? – celtschk Feb 15 '20 at 17:47
  • @celtschk I'm sorry. I do not know anything about curves of degree 5. Where are the exceptions there? – A.G Feb 15 '20 at 17:59
  • @paul garrett I meant that though the number 2 has a lot of special features (think e.g. in your examples, in a lot of theorems on curves that are not valid for varietires of greater dimension, the behaviour of local-global results (such us Hasse-Minkowski) for quadratic forms vs. forms of degree greater than 2, etc.), but I think the question asks for something special about the number 2 as a prime number. You can define varieties of any dimension (not only prime), forms of any order, subgroups of any index, roots of unity of any order, etc. – A.G Feb 15 '20 at 18:13
  • @A.G: I don't know anything about degree 5 curves either. I just noticed that the pattern with degree 2 and degree 3 could go on in two ways. Way 1: degree 2 → characteristic 2 is special. degree 3 → characteristic 2 and 3 are special. degree 5 → characteristic 2, 3 and 5 are special. Way 2: degree 2 → characteristic 2 is special. degree 3 → characteristic 2 and 3 are special. degree 5 → characteristic 2 and 5 are special. With way 1, it would only be the size of characteristic that matters. With way 2, characteristic 2 would be special as the only characteristic that's special in all degrees. – celtschk Feb 15 '20 at 22:20
  • Of course it could also be that there is no pattern at all. Or that the pattern is more complicated. – celtschk Feb 15 '20 at 22:21
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    Could you define $\mathbb Q$ and $SL(n,\mathbb F_q)$? – jvriesem Feb 16 '20 at 02:46
  • @jvriesem $\mathbb Q$ is the set of all rational numbers. It’s Q for Quotient. I was going to ask the same about the other. – gen-ℤ ready to perish Feb 16 '20 at 05:54
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    @jvriesem: $\mathbb F_q$ (where $q$ is prime) is the modulus field of characteristic $q$, that is, the field that you get from $\mathbb Z/q\mathbb Z$. For any field $F$, $SL(n,F)$ is the group of special linear operations on $F^n$, that is, essentially the set of $n\times n$ matrices with entries from $F$ and determinant $1_F$. – celtschk Feb 16 '20 at 07:11
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Parity, either every number is one thing or another, is pretty important.

It's true that $3$ is the only prime divisible by $3$ but some of the primes not divisible by $3$ are $\equiv 1 \pmod 3$ and others are $\equiv -1\pmod 3$ whereas all primes other then $2$ are odd.

If $p<q$ are two different primes then $p+q$ is odd only if $p=2$ but $p+q$ could be any divisibility of $3$. ($3|p+q$ if $p\ne 3$ and $p\equiv -q\pmod 3$. $p+q\equiv 1$ if $p=3$ and $q\equiv 1$ or if $p\equiv q\equiv -1\pmod 3$ and $p+q\equiv -1$ if $p=3$ and $q\equiv 1$ or if $p\equiv q\equiv 1\pmod 3$).

And for $m\le n$ then $p^{m} + p^n = p^m(1+p^n)$ so $p^{m+1}\not \mid p^{m} + p^{n}$ should be a valid result. But... if $p=2$ and $m=n$ then....

Shaun
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fleablood
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    $p\not \mid p^m+p^n$?? – Keith Backman Feb 15 '20 at 18:47
  • Wow. Now that is a brain fart! – fleablood Feb 15 '20 at 19:03
  • I don't see how any of this suggests that $p=2$ is special in general. All these properties have an inherent association with $2$ (e.g., parity or $2$ possible residues modulo $2$) that does not preclude relating another prime $p$ to similar properties associated with $p$. To put it another way, how is "odd vs. even" any more natural than "residue of $0$ vs. $1$ vs. $2$ modulo $3$"? – Jam Feb 19 '20 at 23:56
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    Dichotomies are naturally pertinent. $0, 1, 2$ is three options; you can be one, the other, or neither. even/odd is one or the other. The value of that is either self-evident or it isn't. – fleablood Feb 20 '20 at 05:33
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Here is a personal view on the 'oddness' of $2$:

Parity is important in a logically dichotomous universe; anything other than nothing in the universe is either $A$ or not-$A$ (for each categorization $A$ of things). As others have pointed out, this linguistic or logical 'ambience' brings $2$ to the forefront of our thinking about many things, even when its properties as a prime are not truly unique.

But $2$ is really unusual among the primes (for me) because it is the only prime (indeed the only positive integer $n>1$) for which $x^n+y^n=z^n$ has integer solutions. I consider this fact to be bafflingly improbable. Why are there solutions (and infinitely many of them) for only one integer exponent, and given that is the case, why is that exponent $2$, rather than another among the infinitude of possible primes?

Keith Backman
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  • When stating "the only positive integer n > 1" you've apparently, and rather arbitrary, singled out another suitable integer "1". While 1 is surely a standing out positive integer by itself in all general contexts, your example is strictly speaking true for at least two positive integers: 1 and 2. (Or yeah, even here we have 2 as the number of solutions over n here, so 2 is surely must be something about standing out too!) P.S. We can extend it to all non-negative whole numbers to avoid singling out of n=0 too, since there are no solutions for this case anyway. – Van Jone Dec 07 '23 at 10:28
  • And thinking twice again... Restricting to non-negatives is also a sloppy way, as in this case we strip it from some more (negative) n-s resolving this Diophantine. – Van Jone Dec 07 '23 at 10:32
  • The set of "positive integers raised to the power $1$" is just the set of positive integers, whereas the set of "positive integers raised to any non-unit positive integer power" is a subset of the positive integers. To the extent that one is interested in actions that result in some change (i.e. non-identity functions), exponentiating by $1$ is a non-action. – Keith Backman Dec 07 '23 at 14:55
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$2$ indeed is the smallest prime. Another peculiarity is that it supports the "dichotomy" paradigm, that parallels the logical world: true or false, with or without, left or right.

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Here's some background to the answer. A group is an order pair of a set and a binary operation ⋅ from that set to itself satisfying the following properties.

  • ⋅ is associative
  • There is an identity element
  • Every element has an inverse with respect to that operation.

One element is said to be an inverse of the other when their product in both orders is the identity element. The order of a group is its cardinality. Also, 2 integers are said to be congruent modulo a certain integer when their difference is a multiple of that integer. For example, 2 is congruent to 6 modulo 4. This is written $2 \equiv 6 \mod 4$. The remainder of 10 on division by 4 is 2 so we can also write $10 \mod 4 = 2$. Some people would write $6 = 10 \mod 4$ to mean $6 \equiv 10 \mod 4$ but this is technically incorrect because it is saying that 6 is equal to $10 \mod 4$. But $10 \mod 4 = 2$ and $6 \neq 2$ so $6 \neq 10 \mod 4$.

2 has the property that not all powers of it have a cyclic multiplication modulo group. No other prime number has that property. Take any positive integer. Now take the powers of one more than it modulo the cube of the starting integer such as powers of 11 modulo 1,000. Taking powers of 11 modulo 1,000, we get 001, 011, 121, 331, 641, 051, 561, 171, 881, 691, 601. In any base x, after the $x^{th}$ power, the first digit is a 1 when the base is odd and is 1 more than half the base when the base is even. This shows that the numbers 1 more than a multiple of x modulo $x^3$ form a cyclic group when ever x mod 4 is 1, 3, or 4 but not when x mod 4 is 2. This is something special about the property of being an even number. Another property such as the property of being a multiple of 3 doesn't have something special about it like that.

The multiplication modulo group of any product of 2 odd primes is never cyclic. That's because you can always find a number that's 1 more than a multiple of one of those primes and 1 less than a multiple of the other one of those primes. It's square must be 1 more than a multiple of the semiprime. For example, $21^2$ is 1 more than a multiple of 55. It turns out that all this comes from the fact that 2 is the only prime number that's so small that any integer that's not a multiple of it is 1 more than a multiple of it, making the multiplication modulo group of any odd prime have order an even number.

A subgroup of a group is a subset that is a group with the operation of the original group. A coset of the group is one where each member can be gotten by right multiplying any member of the coset by a member of the original subgroup. When the product of any member of one coset by any member of another coset always gets you a member of the same coset, the subgroup is said to be a normal subgroup. We can then form the cosets into a group and call it the quotient group. When ever a subgroup has 2 cosets, it is necessarily a normal subgroup. For any other prime number, it is not always the case that a subgroup with that number of cosets is a normal subgroup.

Timothy
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